There are a large number of applications that rely on uniform sampling/decomposition of a sphere embedded in 3D space. This submission provides a set of functions that can be used to obtain a variety of different sampling patterns and decompositions of the spherical domain (see demo pic).
Here is a brief summary of the main functions contained in this submission:
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'ParticleSampleSphere' : generates an approximately uniform triangular tessellation of a unit sphere by minimizing generalized electrostatic potential energy of a system of charged particles. By default, initializations are based on random sampling of a sphere, but user defined initializations are also permitted. Since the optimization algorithm implemented in this function has O(N^2) complexity, it is not recommended that 'ParticleSampleSphere' be used to optimize configurations of more than 1E3 particles. Resolution of the meshes obtained with this function can be increased to an arbitrary level with 'SubdivideSphericalMesh'.

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'SubdivideSphericalMesh': increases resolution of triangular or quadrilateral spherical meshes. Given a base mesh, its resolution is increased by a sequence of k subdivisions. Suppose that No is the original number of mesh vertices, then the total number of vertices after k subdivisions will be Nk=4^k*No – 2*(4^k–1). This relationship holds for both triangular and quadrilateral meshes.

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'IcosahedronMesh': generates a triangular mesh of an icosahedron. High-quality spherical meshes can be easily obtained by subdividing this base mesh with the 'SubdivideSphericalMesh' function.

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'QuadCubeMesh': generates a quadrilateral mesh of a cube. High-quality spherical meshes can be easily obtained by subdividing this base mesh with the 'SubdivideSphericalMesh' function.

Hi Anton, Thank you for your soon reply. You are right I did not know that result of triangulation grows exponentially with number of dimensions (n), as you nicely illustrated with the random 100 points, so just for n=20 ‘convhulln’ will return an array of doubles with roughly 8.8E+12 elements and to store that I’ll need 65 400 GB of RAM. I only have 125 GB ☺. What is interesting here is that I do not get matlab ‘out of memory’ error but instead ‘The data is degenerate in at least one dimension – ND set of points lying in (N+1)D space’ . The good news is that I don’t really need to triangulate the points (I don’t need to know indices of the points that comprise the facets of the convex hull) even though it could be used to make sense of the data by sampling some areas and comparing them for instance. Well this takes the function ‘convhulln’ out of the game.
But, how about your code? Why are you amazed it worked for 525D? It basically addresses an optimization problem in 3D that can be for sure addressed in higher dimensions too (perhaps using a more sophisticated scheme like Conjugate gradient, etc). Well the thing is that you have done it and shared it, which all of us appreciate, and I am trying to expand its applicability.
Coming back to the hyper sphere in 525D, I don’t know how such a surface looks like ☺ but I believe that 3 points in a hyper dimensional space will define a triangle (just as they do in 2D and in 3D) so approximating the surface of an hypersphere with a set of triangles makes sense to me, therefore I wonder what do you mean by: “… 525 dimensional convex hull is a terribly poor approximation to a hypersphere…”
Once more thank you for your comments and no, I am not kidding!

Hi Anton. Thank you for sharing your code. I wonder whether you have ever expanded your code to deal with higher dimensions. I need to do sampling on the hypersphere surface (525 – 1029 dimensions) and I accomplish that by normalizing the points draw from the Gaussian distribution, which gives a nice uniform distribution of points on the surface of the hipersphere. Nevertheless I need to refine the sampling so to avoid points laying too close to its neighbors and here is where triangular tessellation sounds just perfect.
I have bypass the 3D checking in your code and it seems to work fine for 525D but it fails when calling ‘convhulln’ function so no triangulation is done therefore I can no make sense of the new point distribution (what you call V).
Do you think ‘convhulln’ is not capable of dealing with 525D or your code ( ParticleSampleSphere ) simply can not be extended to higher dimensions?

Hi Sun, centroids of the spherical triangles can be estimated by taking the average of the three vertices and then projecting the resulting point on the sphere. For example, if x1, x2 and x3 are the vertices of the spherical triangle, you can estimate the centroid as c=(x1+x2+x3)/norm(x1+x2+x3)

One note. The "stratified" option in RandSampleSphere is not really stratified. In fact it is the same as the "uniform" option: a uniform sampling following "Spherical Sampling by Archimedes’ Theorem" [Shao & Badler 96]

Your code is very nice and help me very much. But i need to create point in ellipsoid and I am confusing how to do it. Could you tell how to do it.
Many thanks.

Updates

05 Jun 2012

fixed minor bugs and updated description

23 May 2013

Updated 'RandSamplSphere', the function used to obtain stratified sampling of the unit sphere prior to optimization.

26 Mar 2015

- Added a function that performs sampling of a unit sphere along a spiral
- Updated 'SubdivideSphericalMesh' to work with quad meshes
- Added 'QuadCubeMesh' function to enable spherical decomposition with a quad mesh